Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpglem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
mapdpglem.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
mapdpglem.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
mapdpglem.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
mapdpglem.s |
⊢ − = ( -g ‘ 𝑈 ) |
6 |
|
mapdpglem.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
7 |
|
mapdpglem.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
mapdpglem.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
mapdpglem.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
10 |
|
mapdpglem.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
11 |
|
mapdpglem1.p |
⊢ ⊕ = ( LSSum ‘ 𝐶 ) |
12 |
|
mapdpglem2.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
13 |
|
mapdpglem3.f |
⊢ 𝐹 = ( Base ‘ 𝐶 ) |
14 |
|
mapdpglem3.te |
⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
15 |
|
mapdpglem3.a |
⊢ 𝐴 = ( Scalar ‘ 𝑈 ) |
16 |
|
mapdpglem3.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
17 |
|
mapdpglem3.t |
⊢ · = ( ·𝑠 ‘ 𝐶 ) |
18 |
|
mapdpglem3.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
19 |
|
mapdpglem3.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
20 |
|
mapdpglem3.e |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
21 |
20
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
22 |
14 21
|
eleqtrd |
⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
23 |
|
r19.41v |
⊢ ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
24 |
1 7 8
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
25 |
|
eqid |
⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 ) |
26 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) ) |
27 |
25 26 13 17 12
|
lspsnel |
⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) ) |
28 |
24 19 27
|
syl2anc |
⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) ) |
29 |
1 3 15 16 7 25 26 8
|
lcdsbase |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 ) |
30 |
29
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) ) |
31 |
28 30
|
bitrd |
⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) ) |
32 |
31
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) ) |
33 |
23 32
|
bitr4id |
⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) ) |
34 |
33
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) ) |
35 |
|
df-rex |
⊢ ( ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
36 |
34 35
|
bitr4di |
⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
37 |
|
eqid |
⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 ) |
38 |
37
|
lsssssubg |
⊢ ( 𝐶 ∈ LMod → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) ) |
39 |
24 38
|
syl |
⊢ ( 𝜑 → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) ) |
40 |
13 37 12
|
lspsncl |
⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) ) |
41 |
24 19 40
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) ) |
42 |
39 41
|
sseldd |
⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( SubGrp ‘ 𝐶 ) ) |
43 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
44 |
1 3 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
45 |
4 43 6
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
46 |
44 10 45
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
47 |
1 2 3 43 7 37 8 46
|
mapdcl2 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ) |
48 |
39 47
|
sseldd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝐶 ) ) |
49 |
18 11 42 48
|
lsmelvalm |
⊢ ( 𝜑 → ( 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
50 |
36 49
|
bitr4d |
⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ) |
51 |
22 50
|
mpbird |
⊢ ( 𝜑 → ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
52 |
|
ovex |
⊢ ( 𝑔 · 𝐺 ) ∈ V |
53 |
|
oveq1 |
⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑤 𝑅 𝑧 ) = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |
54 |
53
|
eqeq2d |
⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) ) |
55 |
54
|
rexbidv |
⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) ) |
56 |
52 55
|
ceqsexv |
⊢ ( ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |
57 |
56
|
rexbii |
⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |
58 |
|
rexcom4 |
⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
59 |
57 58
|
bitr3i |
⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) |
60 |
51 59
|
sylibr |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |